Abstract

We consider the reconstruction of a complex-valued object that vibrates in some out-of-plane modes. The reconstruction is based on the phase-retrieval method with the use of two intensity measurements: the two time-averaged image intensities of the object illuminated coherently, which are modulated in two Fourier-transform planes of the object by the use of two filters with exponentially decaying transmittances that complement each other. We discuss the necessary condition of the vibration for the reconstruction method. Computer-simulated examples of retrieving the phases of one-dimensional objects demonstrate that the reconstruction of a sinusoidal-vibrating and a Gaussian random-vibrating object can be treated by this method.

© 1996 Optical Society of America

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References

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  1. H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–39.
    [CrossRef]
  2. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  3. G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
    [CrossRef]
  4. A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.
  5. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.
  6. N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Hague, 1989).
  7. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  8. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  9. J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
    [CrossRef]
  10. J. G. Walker, “Computer simulation of a method for object reconstruction from stellar speckle interferometry data,” Appl. Opt. 21, 3132–3137 (1982).
    [CrossRef] [PubMed]
  11. N. Nakajima, “Phase retrieval from two intensity measurements using the Fourier series expansion,” J. Opt. Soc. Am. A 4, 154–158 (1987).
    [CrossRef]
  12. N. Nakajima, “Phase retrieval using the logarithmic Hilbert transform and the Fourier-series expansion,” J. Opt. Soc. Am. A 5, 257–262 (1988).
    [CrossRef]
  13. N. Nakajima, “Reconstruction of a real function from its Hartley-transform intensity,” J. Opt. Soc. Am. A 5, 858–863 (1988).
    [CrossRef]
  14. N. Nakajima, B. E. A. Saleh, “Reconstruction of a complex-valued object in coherent imaging through a random phase screen,” Appl. Opt. 33, 821–828 (1994).
    [CrossRef] [PubMed]
  15. N. Nakajima, “Two-dimensional phase retrieval by exponential filtering,” Appl. Opt. 28, 1489–1493 (1989).
    [CrossRef] [PubMed]

1994 (1)

1989 (1)

1988 (2)

1987 (1)

1982 (2)

1981 (1)

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Ferwerda, H. A.

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–39.
[CrossRef]

Fiddy, M. A.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Hurt, N. E.

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Hague, 1989).

Levi, A.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

Nakajima, N.

Nieto-Vesperinas, M.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

Ross, G.

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

Saleh, B. E. A.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Stark, H.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

Walker, J. G.

J. G. Walker, “Computer simulation of a method for object reconstruction from stellar speckle interferometry data,” Appl. Opt. 21, 3132–3137 (1982).
[CrossRef] [PubMed]

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

J. G. Walker, “The phase retrieval problem: a solution based on zero location by exponential apodization,” Opt. Acta 28, 735–738 (1981).
[CrossRef]

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Other (6)

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 13–39.
[CrossRef]

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

G. Ross, M. A. Fiddy, M. Nieto-Vesperinas, “The inverse scattering problem in structural determinations,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 15–71.
[CrossRef]

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

N. E. Hurt, Phase Retrieval and Zero Crossings (Kluwer, Dordrecht, The Hague, 1989).

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the geometry of the reconstruction of a vibrating object by measurement of the two image intensities modulated in the two Fourier planes by exponential filters that are the complement of each other.

Fig. 2
Fig. 2

Examples of the reconstruction of the phase of an object vibrating in a sinusoidal mode: (a) the solid and dashed curves represent the original object phase ϕ(x) and the vibrational amplitude of the value of α(x) = π cos(2π0.2x), respectively, when the modulus of the object was set to be constant, (b) the time-averaged Fourier modulus (solid curve) of the vibrating object and the Fourier modulus (dashed curve) of the static object, (c) the time-averaged intensity in one image plane of the optical system shown in Fig. 1, (d) the time-averaged intensity in the second image plane (Fig. 1), (e) the reconstructed object phase, and (f) the absolute value of the first derivative of α(x). The curves shown in (e) and (f) were calculated from the intensities shown in (c) and (d) by the use of.

Fig. 3
Fig. 3

Dependence of the rms reconstruction error (ER) on the noise in the measurement of the sinusoidal-vibration case, in which the effect of noise on the reconstruction is represented by the average SNR ¯ of Eq. (41). Each simulated error was averaged over 10 object phases of identical standard deviation (0.5π) and over the 10 initial seeds used for generating the noise.

Fig. 4
Fig. 4

Dependence of the rms reconstruction error (ER) on the standard deviation σ of the zero-mean Gaussian random vibration for two values of its correlation length, τ = 18.1, 90.5 pixels.

Equations (45)

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G ( u ; t ) = - f ( x 0 ) exp [ i ψ ( x 0 ; t ) ] exp ( - 2 π i u x 0 λ f 1 ) d x 0 ,
g 1 ( x i ; t ) = - G ( u ; t ) exp ( - 2 π c u ) exp ( - 2 π i x i u λ f 1 ) d u ,
g 2 ( x i ; t ) = - G ( u ; t ) exp ( 2 π c u ) exp ( - 2 π i x i u λ f 1 ) d u ,
g 1 ( x i ; t ) = f ( - x i + i λ f 1 c ) exp [ i ψ ( - x i + i λ f 1 c ; t ) ] ,
g 2 ( x i ; t ) = f ( - x i - i λ f 1 c ) exp [ i ψ ( - x i - i λ f 1 c ; t ) ] ,
g ˜ 1 ( x ˜ i ; t ) = f ˜ ( x ˜ i - i c ) exp [ i ψ ˜ ( x ˜ i - i c ; t ) ] ,
g ˜ 2 ( x ˜ i ; t ) = f ˜ ( x ˜ i + i c ) exp [ i ψ ˜ ( x ˜ i + i c ; t ) ] ,
g ˜ 1 ( x ˜ i ; t ) = g 1 ( λ f 1 x ˜ i ; t ) , g ˜ 2 ( x ˜ i ; t ) = g 2 ( λ f 1 x ˜ i ; t ) , f ˜ ( x ˜ i ) = f ( - λ f 1 x ˜ i ) , ψ ˜ ( x ˜ i ; t ) = ψ ( - λ f 1 x ˜ i ; t ) , x ˜ i = x i λ f 1 .
I 1 ( x ˜ i ) = g ˜ 1 ( x ˜ i ; t ) 2 ¯ = f ˜ ( x ˜ i - i c ) 2 A ( x ˜ i - i c ) ,
I 2 ( x ˜ i ) = g ˜ 2 ( x ˜ i ; t ) 2 ¯ = f ˜ ( x ˜ i + i c ) 2 A ( x ˜ i + i c ) ,
A ( x ˜ i ± i c ) = lim T 1 T 0 T exp [ i ψ ˜ ( x ˜ i ± i c ; t ) ] 2 d t ,
f ˜ ( x ˜ i ) = M ( x ˜ i ) exp [ i ϕ ( x ˜ i ) ] .
I 1 ( x ˜ i ) = M ( x ˜ i - i c ) 2 exp [ - 2 Im ϕ ( x ˜ i - i c ) ] × A ( x ˜ i - i c ) ,
I 2 ( x ˜ i ) = M ( x ˜ i + i c ) 2 exp [ - 2 Im ϕ ( x ˜ i + i c ) ] × A ( x ˜ i + i c ) ,
M ( x ˜ i ± i c ) = - [ - M ( x ˜ i ) exp ( 2 π i u x ˜ i ) d x ˜ i ] × exp ( ± 2 π c u ) × exp ( - 2 π i x ˜ i u ) d u .
M * ( x ˜ i + i c ) = M ( x ˜ i - i c ) ,
Im ϕ ( x ˜ i - i c ) = - c d ϕ ( x ˜ i ) d x ˜ i + c 3 3 ! d 3 ϕ ( x ˜ i ) d x ˜ i 3 - c 5 5 ! d 5 ϕ ( x ˜ i ) d x ˜ i 5 + + .
Im ϕ ( x ˜ i - i c ) = - Im ϕ ( x ˜ i + i c ) ,
A ( x ˜ i - i c ) = A ( x ˜ i + i c )
I 1 ( x ˜ i ) I 2 ( x ˜ i ) = exp [ - 4 Im ϕ ( x ˜ i - i c ) ] ,
ϕ ( x ˜ i ) n = 1 N ( a n cos n π l x ˜ i + b n sin n π l x ˜ i ) ,
1 4 ln [ I 1 ( x ˜ i ) I 2 ( x ˜ i ) ] n = 1 N ( - a n sin n π l x ˜ i + b n cos n π l x ˜ i ) × sinh ( n π l c ) .
1 4 ln [ I 1 ( x ˜ i ) I 2 ( x ˜ i ) P ] = - [ Im ϕ ( x ˜ i - i c ) + 1 4 ln P ] .
ψ ( x 0 ; t ) = ( 2 π / λ ) j = 1 J α j ( x 0 ) sin ( ω j t + θ j ) ,
A ( x ˜ i + i c ) = lim T 1 T 0 T | exp [ i 2 π λ j = 1 J α ˜ j ( x ˜ i + i c ) × sin ( ω j t + θ j ) ] | 2 d t , = lim T 1 T 0 T exp { - 4 π λ j = 1 J [ Im α ˜ j ( x ˜ i + i c ) ] × sin ( ω j t + θ j ) } d t ,
A ( x ˜ i + i c ) = lim T 1 T 0 T [ 1 - 4 π λ j = 1 J × [ Im α ˜ j ( x ˜ i + i c ) ] sin ( ω j t + θ j ) + 1 2 ! { - 4 π λ j = 1 J [ Im α ˜ j ( x ˜ i + i c ) ] × sin ( ω j t + θ j } 2 + + ] d t .
A ( x ˜ i + i c ) = 1 + j = 1 J [ 2 π λ Im α ˜ j ( x ˜ i + i c ) ] 2 + 1 4 j = 1 J [ 2 π λ Im α ˜ j ( x ˜ i + i c ) ] 4 + + .
Im α ˜ j ( x ˜ + i c ) = c d α ˜ ( x ˜ i ) d x ˜ i - c 3 3 ! d 3 α ˜ j ( x ˜ i ) d x ˜ i 3 + c 5 5 ! d 5 α ˜ j ( x ˜ i ) d x ˜ i 5 - - .
Im α ˜ j ( x ˜ i + i c ) = - Im α ˜ j ( x ˜ i - i c ) .
A ( x ˜ i + i c ) 1 + ( 2 π λ ) 2 c 2 [ d α ˜ ( x ˜ i ) d x ˜ i ] 2 ,
A ( x ˜ i + i c ) = [ I 1 ( x ˜ i ) I 2 ( x ˜ i ) ] 1 / 2 M ( x ˜ i + i c ) 2 ,
| d α ˜ ( x ˜ i ) d x ˜ i | λ 2 π c { [ I 1 ( x ˜ i ) I 2 ( x ˜ i ) ] 1 / 2 M ( x ˜ i + i c ) 2 - 1 } 1 / 2 .
ψ ( x 0 ; t ) = 2 π λ R ( x 0 ; t ) ,
A ( x ˜ i + i c ) = lim T 1 T 0 T exp [ - 4 π λ Im R ˜ ( x ˜ i + i c ; t ) ] d t = exp [ - 4 π λ Im R ˜ ( x ˜ i + i c ) ] ,
Im R ˜ ( x ˜ i + i c ) = c D ( 1 ) - c 3 3 ! D ( 3 ) + c 5 5 ! D ( 5 ) - - ,
D ( n ) = d n R ˜ ( x ˜ i ) d x ˜ i n .
d n R ˜ ( x ˜ i ) d x ˜ i n = d n R ˜ ( x ˜ i ) d x ˜ i n = 0.
[ Im R ˜ ( x ˜ i + i c ) ] 2 n + 1 = 0 ,             ( n = 0 , 1 , 2 , , ) .
exp [ - 4 π λ Im R ˜ ( x ˜ i + i c ) ] = 1 + 8 π 2 λ 2 [ Im R ˜ ( x ˜ i + i c ) ] 2 + 32 π 4 3 λ 4 [ Im R ˜ ( x ˜ i + i c ) ] 4 + + .
SNR k ( x ) = I n k ( x ) { Var [ I n k ( x ) ] } 1 / 2 ,
SNR ( x ) = ln [ I 1 ( x ) / I 2 ( x ) ] { [ SNR 1 ( x ) ] - 2 + [ SNR 2 ( x ) ] - 2 } 1 / 2 .
SNR ¯ = 1 N n = 1 N SNR ( x n ) .
ER = Min α [ x ϕ r ( x ) + α - ϕ ( x ) 2 x ϕ ( x ) 2 ] 1 / 2 ,
exp [ - 4 π λ Im R ˜ ( x ˜ i + i c ) ] exp [ 1 2 ( 4 π λ ) 2 c 2 σ x 2 ] ,
SNR 1 , 2 ( x ) = M { exp [ ( 4 π λ ) 2 c 2 σ x 2 ] - 1 } - 1 / 2 ,

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